Water saturation equations: Archie, Simandoux, Indonesia, and Fertl

The Archie equation was the first empirical model built (1942) to estimate the water saturation in
non conductive matrix rocks. It usually works well with clean
clastic sandstones and carbonate rocks. Typical parameters for the Archie equation
for consolidated sandstones are a=0.81 (tortuosity), m=1.7 (cementation exponent),
and n=2.0 (saturation exponent). For carbonates, typical
parameters are a=1.0; m=n=2.0.

When the rock matrix has some electrical conductivity, the resistivity is not only
a function of the water resistivity Rw through their free dissolved ions,
but also depends upon the matrix rock minerals beside the non conductive quartz and
calcite matrix grains. The most common cases happen on clastic shaly rocks with
important content of clay minerals.

In these shaly rocks the Archie law over-estimates the water saturation.
Many models consider the Shale Volume (Vshale or Volume of Shale) in the matrix to account for
the excess of conductivity. The Simandoux equation (1963) is among the most used ones.
It reduces mathematically to the Archie equation when n=2 and Vsh=0.
Below are the expressions for the general Archie and Simandoux equations:

Other popular models that deal with shaly sands are the Fertl (1975) equation, and the
Poupon-Leveaux (Indonesia) equation. The Indonesia equation may
work well with fresh formation water. The parameter Rshale
(resistivity of shale) is usually taken from the resistivity reading of a nearby
pure shale, assuming that the clay cements & silt, and the shale nature, are
similar to those of the shaly sand.

The Fertl (1975) equation for shaly sands has the advantage that
does not depend upon Rshale.
It uses instead a reservoir dependent empirically
adjusted 0.25 ≤ α ≤ 0.35 parameter α:

All water saturation equations yield similar results to the Archie equation
for moderately clean sands (see the aqua color clean sand zone
in the picture below). However, the results differ in the case of shaly sand
(see the pink color shaly sand zone), where the Archie law clearly
over-estimates the water saturation (too much water, so a pay zone could be easily
missed if the Archie equation is the only model used).

The figure below compares the results of Archie, Simandoux, Indonesia, and Fertl models

The figure below shows the panel to compute water saturation using Archie, Fertl, & Simandoux equations

The figure below shows the calculation of water saturation using Poupon-Leveaux (Indonesia) & Dual Water Model algorithms